My Favorite One-Liners: Part 24

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

Here’s a problem that could appear in my class in probability or statistics:

Let f(x) = 3x^2 be a probability density function for 0 \le x \le 1. Find F(x) = P(X \le x), the cumulative distribution function of X.

A student’s first reaction might be to set up the integral as

\displaystyle \int_0^x 3x^2 \, dx

The problem with this set-up, of course, is that the letter x has already been reserved as the right endpoint for this definite integral. Therefore, inside the integral, we should choose any other letter — just not x — as the dummy variable.

Which sets up my one-liner: “In the words of the great philosopher Jean-Luc Picard: Plenty of letters left in the alphabet.”

We then write the integral as something like

\displaystyle \int_0^x 3t^2 \, dt

and then get on with the business of finding F(x).

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